Theorem xpdisj1 5035

Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Assertion

Ref	Expression
xpdisj1	|- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Proof of Theorem xpdisj1

Step	Hyp	Ref	Expression
1		inxp 4745	. 2 |- ( ( A X. C ) i^i ( B X. D ) ) = ( ( A i^i B ) X. ( C i^i D ) )
2		xpeq1 4625	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = ( (/) X. ( C i^i D ) ) )
3		0xp 4691	. . 3 |- ( (/) X. ( C i^i D ) ) = (/)
4	2, 3	eqtrdi 2219	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i B ) X. ( C i^i D ) ) = (/) )
5	1, 4	eqtrid 2215	1 |- ( ( A i^i B ) = (/) -> ( ( A X. C ) i^i ( B X. D ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1348   i^i cin 3120   (/)c0 3414   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  djudisj  5038  xp01disjl  6413  xpfi  6907  djuinr  7040